Dynamics of the Mixmaster Universe in a non-commutative Generalized Uncertainty Principle framework

TL;DR

This paper explores how non-commutative generalized uncertainty principles (GUP) influence the chaotic dynamics of the Mixmaster universe, revealing a transition from chaos to quasi-periodic oscillations near the cosmological singularity.

Contribution

It introduces a classical formulation of non-commutative GUP effects on Mixmaster dynamics and derives the modified BKL map, showing a shift from chaos to quasi-periodic behavior.

Findings

Chaotic BKL map replaced by quasi-periodic orbits

Oscillatory behavior of universe's scale factors

Dependence on initial conditions

Abstract

In this work, we examine the dynamical aspects of the cosmological Mixmaster model within the framework of non-commutative generalized uncertainty principle (GUP) theories. The theory is formulated classically by introducing a well-defined symplectic form that differs from the ordinary one, thereby inducing a general deformation of the Poisson brackets describing a precise class of GUP theories. In this general setting, we first investigate the behavior of the Bianchi I and Bianchi II models using Misner variables. Then, we study the Bianchi IX model in the Mixmaster approximation, which is well-known for accurately reproducing the dynamics of the point-particle Universe approaching the cosmological singularity. We derive the corresponding Belinsky-Khalatnikov-Lifshitz (BKL) map and then, by selecting a specific GUP model associated with string theory, we explicitly investigate its resulting features shaped by the non-commutative GUP scheme. Our findings reveal that the chaotic and ergodic behavior typically observed in the standard BKL map, which characterizes the point-Universe's approach to the singularity, is replaced by quasi-periodic orbits in the parameter space of the theory. This corresponds to an oscillatory behavior of the Universe's scale factors, dependent on the initial conditions.